What is the area of triangle ABC above? (1) DBC is an equilateral tri

deshwalankit04 see highlighted part ; how and why ? can you please explain in detail
also ABD is an isosceles triangle not what you have mentioned..

1)B is shown as 90 degrees in ABC and in DBC all will be 60 degrees as it is equlateral;Now we know in ABC ,B is 90 and C is 60 so A is 30.
2)I read it as ABC

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The answer is A
Because from statement 1 we know the breakup of the BDC thus ABC is a 90-60-30 triangle
Statement 2 insufficient we just know ABD is isoceles we Do not know the breakup of any angle
Therefore A is the answer

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This is a moderately difficult question on Geometry, specifically on properties of triangles in general and right angled triangles in particular.

ABC is a right angled triangle, right angled at B. The area of triangle ABC = ½ * AB * BC.
AB = 10√3, so, any data that can help us calculate BC will be sufficient data.

Using Statement I alone, since triangle DBC is an equilateral triangle, angle BCA = 60 degrees. Therefore, in triangle ABC, angle BAC = 30 degrees. So, triangle ABC is a 30-60-90 right triangle.

In a 30-60-90 right triangle, if the side opposite to the 30 degree angle is ‘x’, then the side opposite to the 60 degree angle will be ‘x√3’ and the hypotenuse will be ‘2x’.

In triangle ABC, side AB is opposite 60 degrees. Hence, the side opposite to 30 degrees, BC has to be 10. This is sufficient data to calculate a unique value of the area of the triangle. So, the possible answers are A or D. Options B, C and E can be ruled out.

Using statement II alone, we know that triangle ABD is an isosceles triangle. However we do not know which sides of ABD are equal. Also, even if we know the lengths of the sides, we need data about the angles, to be able to calculate the length of BC. So, statement II alone is insufficient and hence option D can be eliminated.
The correct answer option is A.

If you are conversant with the basic properties of triangles and also specific details like properties of a 30-60-90 triangle, this question should not take you more than a minute and a half to solve. The second statement, though, should be interpreted with care. When it comes to DS questions on Geometry, always remember that unless you get a UNIQUE figure from the data given, the data cannot be considered as sufficient. And in our case, statement II will not give us a unique figure.

Hope this helps!